NAG Library Function Document

nag_sparse_sym_sol (f11jec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

+− 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

nag_sparse_sym_sol (f11jec) solves a real sparse symmetric system of linear equations, represented in symmetric coordinate storage format, using a conjugate gradient or Lanczos method, without preconditioning, with Jacobi or with SSOR preconditioning.

2 Specification

#include <nag.h>

#include <nagf11.h>

void

nag_sparse_sym_sol (Nag_SparseSym_Method method, Nag_SparseSym_PrecType precon, Integer n, Integer nnz, const double a[], const Integer irow[], const Integer icol[], double omega, const double b[], double tol, Integer maxitn, double x[], double *rnorm, Integer *itn, Nag_Sparse_Comm *comm, NagError *fail)

3 Description

nag_sparse_sym_sol (f11jec) solves a real sparse symmetric linear system of equations:

A x = b,

using a preconditioned conjugate gradient method (see Barrett et al. (1994)), or a preconditioned Lanczos method based on the algorithm SYMMLQ (Paige and Saunders (1975)). The conjugate gradient method is more efficient if

A

is positive definite, but may fail to converge for indefinite matrices. In this case the Lanczos method should be used instead. For further details see Barrett et al. (1994).

The function allows the following choices for the preconditioner:

no preconditioning;
Jacobi preconditioning (see Young (1971);
symmetric successive-over-relaxation (SSOR) preconditioning (see Young (1971)).

For incomplete Cholesky (IC) preconditioning see nag_sparse_sym_chol_sol (f11jcc).

The matrix

A

is represented in symmetric coordinate storage (SCS) format (see the f11 Chapter Introduction) in the arrays a, irow and icol. The array a holds the nonzero entries in the lower triangular part of the matrix, while irow and icol hold the corresponding row and column indices.

4 References

Barrett R, Berry M, Chan T F, Demmel J, Donato J, Dongarra J, Eijkhout V, Pozo R, Romine C and Van der Vorst H (1994) Templates for the Solution of Linear Systems: Building Blocks for Iterative Methods SIAM, Philadelphia

Paige C C and Saunders M A (1975) Solution of sparse indefinite systems of linear equations SIAM J. Numer. Anal. 12 617–629

Young D (1971) Iterative Solution of Large Linear Systems Academic Press, New York

5 Arguments

1: method – Nag_SparseSym_MethodInput

On entry: specifies the iterative method to be used.

$method = Nag_SparseSym_CG$: The conjugate gradient method is used.
$method = Nag_SparseSym_Lanczos$: The Lanczos method (SYMMLQ) is used.

Constraint:

method = Nag_SparseSym_CG

Nag_SparseSym_Lanczos

2: precon – Nag_SparseSym_PrecTypeInput

On entry: specifies the type of preconditioning to be used.

$precon = Nag_SparseSym_NoPrec$: No preconditioning is used.
$precon = Nag_SparseSym_SSORPrec$: Symmetric successive-over-relaxation is used.
$precon = Nag_SparseSym_JacPrec$: Jacobi preconditioning is used.

Constraint:

precon = Nag_SparseSym_NoPrec

Nag_SparseSym_SSORPrec

Nag_SparseSym_JacPrec

3: n – IntegerInput

On entry: the order of the matrix

A

Constraint:

n \geq 1

4: nnz – IntegerInput

On entry: the number of nonzero elements in the lower triangular part of the matrix

A

Constraint:

1 \leq nnz \leq n \times (n + 1) / 2

5: a[nnz] – const doubleInput

On entry: the nonzero elements of the lower triangular part of the matrix

A

, ordered by increasing row index, and by increasing column index within each row. Multiple entries for the same row and column indices are not permitted. The function nag_sparse_sym_sort (f11zbc) may be used to order the elements in this way.

6: irow[nnz] – const IntegerInput

7: icol[nnz] – const IntegerInput

On entry: the row and column indices of the nonzero elements supplied in

A

Constraints:

irow and icol must satisfy the following constraints (which may be imposed by a call to nag_sparse_sym_sort (f11zbc)):;
$1 \leq irow [i] \leq n$ and $1 \leq icol [i] \leq irow [i]$ , for $i = 0, 1, \dots, nnz - 1$ ;
$irow [i - 1] < irow [i]$ or $irow [i - 1] = irow [i]$ and $icol [i - 1] < icol [i]$ , for $i = 1, 2, \dots, nnz - 1$ .

8: omega – doubleInput

On entry: if

precon = Nag_SparseSym_SSORPrec

, omega is the relaxation argument

ω

to be used in the SSOR method. Otherwise omega need not be initialized.

Constraint:

0.0 \leq omega \leq 2.0

9: b[n] – const doubleInput

On entry: the right-hand side vector

b

10: tol – doubleInput

On entry: the required tolerance. Let

x_{k}

denote the approximate solution at iteration

k

, and

r_{k}

the corresponding residual. The algorithm is considered to have converged at iteration

k

if:

{‖r_{k}‖}_{\infty} \leq τ \times ({‖b‖}_{\infty} + {‖A‖}_{\infty} {‖x_{k}‖}_{\infty}) .

tol \leq 0.0

τ = \max (\sqrt{ε}, \sqrt{n}, ε)

is used, where

ε

is the machine precision. Otherwise

τ = \max (tol, 10 ε, \sqrt{n}, ε)

is used.

Constraint:

tol < 1.0

11: maxitn – IntegerInput

On entry: the maximum number of iterations allowed.

Constraint:

maxitn \geq 1

12: x[n] – doubleInput/Output

On entry: an initial approximation of the solution vector

x

On exit: an improved approximation to the solution vector

x

13: rnorm – double *Output

On exit: the final value of the residual norm

{‖r_{k}‖}_{\infty}

, where

k

is the output value of itn.

14: itn – Integer *Output

On exit: the number of iterations carried out.

15: comm – Nag_Sparse_Comm *Input/Output

On entry/exit: a pointer to a structure of type Nag_Sparse_Comm whose members are used by the iterative solver.

16: fail – NagError *Input/Output

The NAG error argument (see Section 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ACC_LIMIT

The required accuracy could not be obtained. However, a reasonable accuracy has been obtained and further iterations cannot improve the result.

NE_ALLOC_FAIL

Dynamic memory allocation failed.

NE_BAD_PARAM

On entry, argument method had an illegal value.

On entry, argument precon had an illegal value.

NE_COEFF_NOT_POS_DEF

The matrix of coefficients appears not to be positive definite (conjugate gradient method only).

NE_INT_2

On entry,

nnz = ⟨value⟩

n = ⟨value⟩

.
Constraint:

1 \leq nnz \leq n \times (n + 1) / 2

NE_INT_ARG_LT

On entry,

maxitn = ⟨value⟩

.
Constraint:

maxitn \geq 1

On entry,

n = ⟨value⟩

.
Constraint:

n \geq 1

NE_INTERNAL_ERROR

An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.

NE_NOT_REQ_ACC

The required accuracy has not been obtained in maxitn iterations.

NE_PRECOND_NOT_POS_DEF

The preconditioner appears not to be positive definite.

NE_REAL

On entry,

omega = ⟨value⟩

.
Constraint:

0.0 \leq omega \leq 2.0

NE_REAL_ARG_GE

On entry, tol must not be greater than or equal to 1.0:

tol = ⟨value⟩

NE_SYMM_MATRIX_DUP

A nonzero element has been supplied which does not lie in the lower triangular part of the matrix

A

, is out of order, or has duplicate row and column indices, i.e., one or more of the following constraints has been violated:

$1 \leq irow [i] \leq n$ and $1 \leq icol [i] \leq irow [i]$ , for $i = 0, 1, \dots, nnz - 1$
$irow [i - 1] < irow [i]$ , or
$irow [i - 1] = irow [i]$ and $icol [i - 1] < icol [i]$ , for $i = 1, 2, \dots, nnz - 1$ .

Call nag_sparse_sym_sort (f11zbc) to reorder and sum or remove duplicates.

NE_ZERO_DIAGONAL_ELEM

The matrix

A

has a zero diagonal element. Jacobi and SSOR preconditioners are not appropriate for this problem.

7 Accuracy

On successful termination, the final residual

r_{k} = {b - A x}_{k}

, where

k = itn

, satisfies the termination criterion

{‖r_{k}‖}_{\infty} \leq τ \times ({‖b‖}_{\infty} + {‖A‖}_{\infty} {‖x_{k}‖}_{\infty}) .

The value of the final residual norm is returned in rnorm.

8 Parallelism and Performance

Not applicable.

9 Further Comments

The time taken by nag_sparse_sym_sol (f11jec) for each iteration is roughly proportional to nnz. One iteration with the Lanczos method (SYMMLQ) requires a slightly larger number of operations than one iteration with the conjugate gradient method.

The number of iterations required to achieve a prescribed accuracy cannot be easily determined a priori, as it can depend dramatically on the conditioning and spectrum of the preconditioned matrix of the coefficients

\bar{A} = M^{- 1} A

10 Example

This example program solves a symmetric positive definite system of equations using the conjugate gradient method, with SSOR preconditioning.

NAG Library Function Documentnag_sparse_sym_sol (f11jec)

+− Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG Library Function Document

nag_sparse_sym_sol (f11jec)